Quit[];

sigma[s_]=Sqrt[1-4/s];
Jbar[s_]=1/(16 Pi^2)(sigma[s]Log[(sigma[s]-1)/(sigma[s]+1)]+2);

Limit[Jbar[x],x->0]

Limit[((Jbar[s+I im]-Jbar[s-I im])/(2I)),im->0,Direction->"FromAbove",Assumptions->{s>4}]

Limit[((Jbar[s+I im]-Jbar[s-I im])/(2I)),im->0,Direction->"FromAbove",Assumptions->{s<4}]

Jbar2[s_]=Jbars0+s/Pi Integrate[1/(16 Pi) sigma[x]/(x(x-s)),{x,4,Infinity}]

Jbar3[s_]=Jbars0+(((Sqrt[-(-4+s) s])+(-4+s) ArcCsc[(2/(Sqrt[s]))])/(8 (π^2) (Sqrt[4-s]) (Sqrt[s])));

Jbar[5+0.0001 I] // N

Jbar3[5+0.0001 I] /. Jbars0-> 0 // N